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We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong integrability condition on $V$, using the support of its Fourrier transform. We then use this condition to prove that a real trigonometric polynomial potential is integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of integrable potentials in dimension $2$ and $3$, and recover several integrable cases. These potentials after a complex variable change become real, and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high degree first integrals are explicitly integrated., Comment: 29 pages

We prove that the only meromorphically integrable planar homogeneous potential of degree k <> -2,0,2 having a multiple Darboux point is the potential invariant by rotation. This case is a singular case of the Maciejewski-Przybylska relation on eigenvalues at Darboux points of homogeneous potentials, and needed before a case by case special analysis. The most striking application of this Theorem is the complete classification of integrable real analytic homogeneous potentials in the plane of negative degree., Comment: 22 pages

We present various properties of algebraic potentials, and then prove that some Morales-Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. This allows in particular to precise some non-integrability proofs in celestial mechanics, where the mutual distances between the bodies appear in the potentials, and thus making this analysis unavoidable., Comment: 8 pages

We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface., Comment: 14 pages, 8 references

We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r,\theta)=r^{-1} h(\exp(i\theta)) with h a polynomial of degree less than 3. We find then all meromorphically integrable potentials of this form., Comment: 37 pages, 2 figures; Journal of Mathematical Physics 2012, Volume 53, Issue 8

We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $-1$ which are real analytic on $\mathbb{R}^2\setminus \{0\}$. In the more general case when $V$ is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point $c$ with $V'(c)=-c,\; c_1^2+c_2^2\neq 0$ and $\hbox{Sp}(\nabla^2 V(c)) \subset\{-1,0,2\}$. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case $V'(c)=0$., Comment: 40 pages, 42 references